Supercontinuum generation enabled a series of key technologies such as frequency comb sources, ultrashort pulse sources in the ultraviolet or the mid-infrared, as well as broadband light sources for spectroscopic methods in biophotonics. Recent advances utilizing higher-order modes have shown the potential to boost both bandwidth and modal output distribution of supercontinuum sources. However, the strive towards a breakthrough technology is hampered by the limited control over the intra- and intermodal nonlinear processes in the highly multi-modal silica fibers commonly used. Here, we investigate the ultrafast nonlinear dynamics of soliton-based supercontinuum generation and the associated mode coupling within the first three lowest-order modes of accurately dispersion-engineered liquid-core fibers. By measuring the energy-spectral evolutions and the spatial distributions of the various generated spectral features polarization-resolved, soliton fission and dispersive wave formation are identified as the origins of the nonlinear broadening. Measured results are confirmed by nonlinear simulations taking advantage of the accurate modeling capabilities of the ideal step-index geometry of our liquid-core platform. While operating in the telecommunications domain, our study allows further advances in nonlinear switching in emerging higher-order mode fiber networks as well as novel insights into the sophisticated nonlinear dynamics and broadband light generation in pre-selected polarization states.